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Theorem supsn 7994
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
supsn  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )

Proof of Theorem supsn
StepHypRef Expression
1 dfsn2 4015 . . . 4  |-  { B }  =  { B ,  B }
21supeq1i 7967 . . 3  |-  sup ( { B } ,  A ,  R )  =  sup ( { B ,  B } ,  A ,  R )
3 suppr 7993 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
433anidm23 1323 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
52, 4syl5eq 2482 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
6 ifid 3952 . 2  |-  if ( B R B ,  B ,  B )  =  B
75, 6syl6eq 2486 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ifcif 3915   {csn 4002   {cpr 4004   class class class wbr 4426    Or wor 4774   supcsup 7960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-po 4775  df-so 4776  df-iota 5565  df-riota 6267  df-sup 7962
This theorem is referenced by:  supxrmnf  11603  ramz  14946  xpsdsval  21327  ovolctb  22321  nmoo0  26277  nmop0  27474  nmfn0  27475  esumnul  28708  esum0  28709  ovoliunnfl  31686  voliunnfl  31688  volsupnfl  31689  fourierdlem79  37617  sge0z  37751  sge00  37752
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